Exact uniform modulus of continuity and Chung's LIL for the generalized fractional Brownian motion

Abstract

The generalized fractional Brownian motion (GFBM) $X:=\{X(t)\}_{t\ge0}$ with parameters $\gamma \in [0, 1)$ and $\alpha\in \left(-\frac12+\frac{\gamma}{2}, \, \frac12+\frac{\gamma}{2} \right)$ is a centered Gaussian $H$-self-similar process introduced by Pang and Taqqu (2019) as the scaling limit of power-law shot noise processes, where $H = \alpha-\frac{\gamma}{2}+\frac12 \in(0,1)$. When $\gamma = 0$, $X$ is the ordinary fractional Brownian motion. For $\gamma \in (0, 1)$, GFBM $X$ does not have stationary increments, and its sample path properties such as H\"older continuity, path differentiability/non-differentiability, and the functional law of the iterated logarithm have been investigated recently by Ichiba, Pang and Taqqu (2020). They mainly focused on sample path properties that are described in terms of the self-similarity index $H$ (e.g., LILs at the origin and at infinity). In this paper, we further study the sample path properties of GFBM $X$ and establish the exact uniform modulus of continuity, small ball probabilities, and Chung's law of iterated logarithm. Our results show that the local regularity properties of GFBM $X$ away from the origin are determined by the index $\alpha +\frac1 2$, instead of the self-similarity index $H$. This is in contrast with the properties of ordinary fractional Brownian motion whose local and asymptotic properties are determined by the single index $H$.